Fuchsian group

A Fuchsian group is a mathematical concept describing a set of transformations that act on the hyperbolic plane, preserving its geometric properties. These transformations are like symmetries that map the plane onto itself without distortion, often generated by repeating basic patterns. Fuchsian groups are important in understanding complex structures called Riemann surfaces and appear in areas like geometry, number theory, and physics. They help mathematicians study how shapes and patterns behave in non-Euclidean, hyperbolic spaces, revealing deep insights into the nature of space, symmetry, and mathematical structures.